From time to time, math folks can’t help wrestling with the old, pot-stirring question “Is [*algebra/calculus/trigonometry/mathematics*] class really necessary?”

The argument goes like this: At every step of education, students face math requirements.

What’s weird is that, once you’ve cleared the bar, you rarely *use* the math you learned.

Math is just a gatekeeper, a sorting mechanism, a bouncer used to keep some people out of the party. So why not eliminate those dumb requirements altogether?

In the ensuing debate, math folks leap to defend the discipline. Skeptics parry with counterattacks. In the end, we talk, blog, and Tweet right past each other.

Both sides dislike math education’s competitive, exclusionary nature. One side aims to overcome it by reorienting towards a higher purpose. (“Math is beautiful/useful/the best way to learn reasoning!”) The other side prefers to curtail math’s presence. (“End these foolish requirements!”) But to me, all of this dances around an obvious truth.

Why does math function as a gatekeeper?

Because our educational system is full of gates.

The judges behind these gates find themselves sifting through piles of applications. They turn to math as a simple signal of desirable qualities.

We all know it’s not a perfect indicator. But admissions and HR departments seem to find it useful anyway.

Meanwhile, even when it’s not required, students pursue mathematics in order to prove their diligence, muscle, and intellectual worth in the competitive economy. They want to get through the gates, and they see math as the key.

Mathematicians don’t necessarily encourage or desire this dynamic. It happens above and around them—sometimes even in spite of them.

This is a dreary thought for both the anti- and pro-math camps. The exclusivity comes first, and math simply rises up to fill the gap. Eliminating the gatekeeper won’t increase the number of spots at Harvard or jobs at Google. Students will just seek other grounds on which to compete, and the folks evaluating students will seek other grounds on which to distinguish them.

This frame helps me understand why each side of the algebra debate sounds so out-of-touch to the other. The purpose math-lovers would choose for the subject doesn’t necessarily match the purpose assigned to it.

The idea of “math as competitive platform” discomforts me. Still, I see a modest path forward.

First, frankly acknowledge the system’s competitive nature. We math teachers wield undeniable power over young people’s lives, and we should aim to do so responsibly, openly, and evenhandedly.

Second, just because math plays a role in sorting doesn’t mean it can’t serve eight hundred other magnificent purposes. In spite of the constraints, we should endeavor to make education as rich, meaningful, and “useful” (however you want to define that word) as possible.

Is algebra/calculus/trigonometry “necessary”? Well, apart from maple syrup, very little in life is. But is mathematics bursting with potential to inspire and to enrich students’ mental lives? The answer to that, I believe, is a resounding “yes.”

This post makes me feel conflicted.

“Both sides dislike math education’s competitive, exclusionary nature.”

Math isn’t competitive or exclusionary by nature. Math has been put to use as a gatekeeper in a competitive and sometimes exclusionary world. You acknowledge that, but this sentence bothers me.

From my limited knowledge of the history, Harvard raised the gate. Somewhere in the early part of the 19th century someone said, students who arrive at here with basic math skills do better. Furthermore, it is misuse of limited faculty resources to teach basic algebra, hence all incoming students will be required to have at least one year of algebra! And a whole bunch of colleges followed Harvard’s lead. And then Harvard said Geometry, too!… and some more algebra.

The gates are there because the gates serve a purpose.

Heck, even Plato’s Academy said “Let no-one ignorant of geometry enter here.”

On the “usefulness” of mathematics,

Why is math the only subject that we ask “when will I ever use this?” It doesn’t seem that question is asked of the language / art / social science / science teachers. I would like to tell myself that this is because math is, in fact, so useful that we expect it to be useful.

Aesthetics

Even the very good math students rarely appreciate the beauty and imagination behind the mathematics. The idea that mathematics has beauty and the people “do math” purely for aesthetic reasons seems alien. For too many, the creativity has been drummed out of the subject.

Just tell me how to get the answer.

I was in this camp for a long time. I didn’t need to know how the mean value theorem might be use to prove the fundamental theorem of calculus, just show me the tricks to do the integrations. Because the teacher says so, was a perfectly good explanation. It is for every other class, why not math, too? I was a good monkey who knew which crank to turn to get this big math machine to spit out a prize.

And in the world of competitive testing, winning the prize is goal. We need a good mark on the SAT, and we need to pass the AP exams. And, neither of those will ask me to prove

|A+B| ≤ |A| + |B|.

We learn Babylonian Mathematics (memorization of formula and techniques) while mathematicians want us to learn Greek Mathematics (theorems and proof). And one crowds out the other. However, Babylonian mathematics has a limit. There is only finite capacity to hold onto all of these disparate facts. Greek math provides the connections that ultimately allow us to push much farther.

I see why you’re bothered by that sentence – “nature” was probably a bad word choice. I meant something more like “role” or “state.”

Thanks for sharing that bit of Harvard history – I didn’t know that. From my very superficial reading of other history, I suspect that the decision-makers at Harvard were perhaps boosting an existing signal in the culture, rather than emitting a new one, but it’s hard to say.

I sometimes think that the ills you describe (a lacking sense of aesthetics, an emphasis on procedure) all emanate from this gatekeeper role. If I need to perform Task X to access good jobs, then I’m going to try to break Task X down into a hundred tiny mechanical steps that I can practice ad infinitum. Meanwhile, the gatekeeper setting Task X will feel pressured not to pick an open-ended, creative task, because (A) these are harder to grade, and (B) all the people trying to get through the gate will complain that it’s arbitrary and unfair.

You forgot one other necessary thing…BACON!!!

But yes. No mathematics is necessary. However, most are useful if approached the right way. Unfortunately, a competitive environment is not ideal for fostering a love of mathematics and getting students to see it’s real worth. By the time students reach middle/high school, any inherent “love for learning” has been thoroughly and utterly quenched.

“By the time students reach middle/high school, any inherent “love for learning” has been thoroughly and utterly quenched.”

Funny, since my interest in math started in high school, when they started teaching algebra and geometry. 🙂

FWIW, your line “I want only geniuses at my firm! Bring me mathematicians and physicists” couldn’t help but remind me of Jim Simon’s practice of hiring mathematician/physicist types (instead of financial types) to run his wildly successful Renaissance Technology Fund. Here’s what Wikipedia says of it:

“Renaissance is a firm run by and for scientists, employing preferably those with non-financial backgrounds for quantitative finance research like mathematicians, statisticians, pure and experimental physicists, astronomers, and computer scientists. Wall Street experience is frowned on and a flair for science is prized. It is a widely held belief within Renaissance that the herdlike mentality among business school graduates is to blame for poor investor returns.”

Interesting! I read “The Money Formula” last year and learned a lot about the job/culture of quants. Feels like such a foreign land to me.

Ben

There is an unforeseen positive externality to math being a gatekeeper. Many more students engage with math than they would otherwise. Is this a good thing in and by itself? And Jordan Ellenberg had I think the right idea when he chose the title for his book “How not to be wrong? The power of mathematical thinking”. Our emphasis on compulsory math education produces, in varying degrees, students who will grow up to be less wrong than they would be without their limited engagement with math in school. And that is unequivocally a good thing.

Sunil

Hi Sunil, I appreciate the optimism of that take!

I know that sound students have great, mind-enriching experiences of math education; I had one myself.

Oh the whole, though, I worry that math education doesn’t do much for most students to sharpen their sense of truth and logic. The emphasis on blindly following rote procedures may even undermine one’s critical sense…

A lot of similar ideas to this post:

http://teachlikeachampion.com/blog/utopia-and-other-aristocracies/

What’s worse than using math ability as a gatekeeper? Using race or religion or gender or money or status or political connections.

Yeah, I think that’s true and pithily said. I’m sympathetic that we could do better than we do now, but it worries me when people brush aside the idea that the economy will inevitably have gates and gatekeepers.

Scott Alexander has a good post somewhere where he draws a distinction between the silly argument “Meritocracy, in principle, is a bad idea” and the much more credible argument “Meritocracy, in practice, is functioning poorly.”

I went and found it and it is great too: http://slatestarcodex.com/2017/07/24/targeting-meritocracy/

There is a related argument that I have heard:

“Instead of using Calculus as the gatekeeping math course we should use Linear Algebra because it is more likely to be used in modern data-heavy society.”

Personally, I agree. However the momentum of “We use calculus to sort students” is very hard to overcome.

Interesting – I’ve heard the argument more with probability and statistics than linear algebra, but I guess you can substitute either one.

I think there is one simple move that would actually shift the emphasis towards statistics in the US: collapse the two AP Calculus courses into one. Currently, 50-100k students per year take a second year of calculus rather than a first year of statistics. I conjecture that if you eliminated the second year of calculus, those students would leap over to statistics, so as to keep their AP count up.

As a student who has been through many AP exams, I would just like to say that AP statistics is a sad joke of what it could actually be. One could say this is only my personal opinion, but when universities in Canada and the UK explicitly state that “AP Statistics does not provide the appropriate preparation”, you know there is a problem.

I would like to say that this dumbing-down is the main issue, but I simply don’t know at this point. It saddens me that many of these gate-keeping strategies aren’t used to teach what students will actually face when they enter university or become employed. It shouldn’t be that hard, in the sense that perhaps you could at least teach the 101 version?

This act of forcing education upon people is what I believe breaks the ability to love education. Perhaps it is required at a young age in order to show people the school and the classroom, but do we really need force down people’s throats arbritrary topics so they become “better citizens”? Why not, at a certain age such as 14 or 15, allow more control for the student to choose topics they desire? I find that this lack of choice or control for students is what causes the depression and hate for school (and, consequently, the idea of education) in my country where school only becomes a mechanism to teach students for their university entrance exams and nothing more. (Turkey)

My main hope is creating an AI which could act as a teacher of any topic for people, perhaps even removing the gate-keeping requirement since the AI in itself could act as a university. It could still be used as a gate-keeping strategy as it might cross-check other people for their skill level, given that they give permission for the university or employer to see/use that data.

So many problems, so little time.

this is an interesting way of looking at the whole question of when will I ever use this math in my life? the upper elementary school my sons are in actually divides their homerooms based on math ability which I found interesting since typically elementary schools put more emphasis on reading than math and that flips in middle school and high school. I’m not sure we will ever find a good answer as to what math or other subject should be the gate keeper, but it is an interesting question to ponder

Used to be Latin and Greek that got you into polite society. I don’t know that most people used it daily. Exceptional athletic ability will also get you into a university, but it will almost surely decline by 30 years old. Maybe the kind of signal doesn’t matter that much.

But ability to generate one does. Relevantly, much of the real life work today consist of signaling, for yourself, your team, your company. So, being good at signaling, which means putting the busy bee work of becoming exceptional at *something*, is actually a good predictor of future career success!

Yeah, I think I agree with that. It jibes with my sense that one’s academic background (test scores, GPA, whatever) is useful for evaluating someone fresh out of high school/college, but ought to be a marginal consideration if you’re evaluating someone with an actual career track record (i.e., more recent signals to consider).

I love your work, laugh and cry when reading it regularly, and recommend it to all the math folks I know.

This post, however, leaves me shaking my head. Briefly (and only partly related to this comic – which also had me laughing):

1. I deeply disagree that mathematicians don’t want the privileged place we’re in. We don’t generally recognize it as privilege, but it is. We might say we don’t want it, but underneath we’re desperately afraid to lose it. We are looked up to because we made it through all of those gates. Our self-worth, our position in society, our being looked up to as “smart” or “brilliant” – it’s all part and parcel of exactly the system you’re describing. We don’t dismantle it because it elevates us.

2. Our gatekeeping doesn’t happen in a vacuum; it doesn’t happen in a world where all other things are equal (or all other variables are held constant). In our society, rich school districts (largely white & Asian) pay top wages and attract and keep the best prepared teachers in largely segregated schools. Too many black and brown kids huddle in abysmal conditions (see Baltimore schools during the recent cold snap) with mathematical preparations to match their surroundings. Nearly everywhere, girls get less encouragement, less support, more harassment, and more reasons to flee math and STEM fields in general. Our gatekeeping is enforcing a world order that is not fair, not equitable, and not just. We are part of the system that perpetuates those inequities. If we’re not going to start dismantling it, at least we should admit what we’re doing.

3. Alternatives to the “death march toward Calculus” and “math as beauty and art” are gaining ground all over. The Quantitative Reasoning/Literacy movement aims to provide useful math, especially to those outside of STEM fields. The Math for Social Justice folks want to leverage the teaching of mathematics (especially to the underserved) as a way of upending the current social and economic order. The Alternative Pathways advocates (e.g., Uri Treisman’s Dana Center) are providing new curricula that actually align with students’ future plans – and are seeing huge gains in achievement and persistence.

4. Finally, I’d love for our community to move from treating students largely as future workers (which is what our gatekeeper status is mostly about) to considering them as citizens. What mathematics does an informed citizen need to understand? What role should our community be playing in, as Dan Meyer put it, getting people to believe fewer lies?

Sorry to get carried away here. As I said up front, I love your work and look forward to more of it. This one just rubbed me the wrong way.

Dave

Hey Dave, thanks for reading and responding. I appreciate all four points, and genuinely agree with all of them. You’re right that we math folks cherish our sense of superiority, and need to be called out on it. You’re right that math’s gatekeeping can perpetuate (or even amplify) existing inequities in society. You’re right that math education often fails to embrace students’ role as citizens. And you’re right to highlight the folks working on alternative models (thanks for that – I was only familiar with some of this work).

I think it’s very fair to have been rubbed the wrong way by this post; it’s tackling a reality that rubs *me* the wrong way, and I’m still searching for the right language to talk about it. I guess what I find hard is this: *why* is math education so competitive and exclusive?

If the causes are fundamentally top-down – it’s an intellectual elite feeding its own ego, or a racial hegemony safeguarding its economic control – then the answer is very clear: burn it down! Eliminate the high-stakes tests, ditch the wasteful curricula, and rebuild a more equitable system from scratch.

If the causes are fundamentally bottom-up – it’s actors all across the economy seeing math attainment as a signal of something desirable (intelligence, diligence, some vague capacity for excellence) – then the answer is less clear. Do we seek to move the gatekeeper function to another locus in the educational system, freeing math up to serve other purposes? (That’s the import of the “gradeless learning” approach.) Do we lean into the gatekeeper function, and try to *improve* the signal, making math attainment a *better* measure of qualities we deem valuable? (Not a popular option.) Do we say, “The gatekeeping isn’t the issue, the problem is the inequity,” and then focus our resources and energies on helping the disadvantaged fare better in the competition? (That’s what TFA and many charter schools go for.) Or do we say, “The ‘college for all’ movement is just cranking up the speed on the credential treadmill,” and try to find alternate pathways for disadvantaged students to reach professional and economic stability? (That points towards vocational education.)

I suspect that there are top-down and bottom-up forces at play. My own imperfect eye sees more of the latter. And the post probably does too much to disguise the fact that I genuinely don’t know how to respond.

This is just offensive. Ben, please consider deleting it. (FWIW, the last two paragraphs are lifted from an obscure self-published screed.)

Yeah, would’ve been happy to engage with the argument but the vulgarity seemed like a sign they were venting rather than looking to discuss. Deleted.

Awww….really like this post (in spite of my myself). Need to do my taxes (not my forte) but go Maths!

Math isn’t an arbitrary gatekeeper. It’s an important language and a way of thinking.

Do you really want a doctor who doesn’t understand the difference between an A1C test and a glucose test or can’t apply Bayesian statistics in clinical practice? Hint, the A1C measurement is a time-weighted integral of glucose exposure. Maybe a medical student who doesn’t know calculus or statistics can learn just enough on the fly in medical school, but someone who knows math cold is much more likely to get through medical school and be a good doctor in practice.

There are a lot of fields that require mathematics both in school and in practice. A lot of them involve science and technology, but people in finance or accounting or architecture or city planning also use mathematics and mathematical reasoning all the time. Anyone who hopes to run a small business on anything but the most informal basis needs and uses math. Without a high school math background, a whole lot of majors are next to impossible in college. This isn’t just an arbitrary gateway. If you don’t have the math you need, you either have to be an amazingly quick study, or you are not going to pass the course. If you work in the field, you might not need your math every day, but it’s part of the language and part of he work. Any job that require visualization and estimation require mathematics. Work out a psychiatric care plan and you’ll need to estimate the resources.

Knowing math is like knowing grammar or literary analysis or rhetoric. Sure, it is possible to get through life without it, but actually learning something makes further learning a lot easier. Being able to read a paragraph and answer questions about it isn’t just an arbitrary roadblock that is keeping the next Einstein from a physics PhD. Being able to add two fractions – and being unable to add two fractions is one of the primary reason community college students can’t pass their math courses – isn’t just an arbitrary roadblock that is keeping the next E.M.Forster from writing about the flaws in colonialism.

Maybe you have this view because you are a math teacher. Aside from the subject matter, how much math does a math teacher need? Be honest. It’s not any more than an English teacher, not a whole lot. An English teacher needs a lot more English than a math teacher needs math. In that sense, what you are teaching is arbitrary, but it isn’t arbitrary to your students.

We’ve raised the gate with math during the same interval we’ve lowered one with language.

It seems to me that once upon a time “elementary education” finished with arithmetic, English composition, and English (American) literature. A.k.a Reading, ‘riting, and ‘rithmetic. “Secondary education” introduced Euclid — and Greek and/or Latin.

Pedagogical theorists suggest that a “general” intelligence or aptitude for learning (G) is the vector sum of a math aptitude and a language aptitude. (There is, I think, reason to suggest another dimension of aptitude for social interaction: such that the TV cariacture character Sheldon Cooper is considered a “genius” for being very long on math, pretty long on language, but ludicrously short in social awareness. There are those gifted with very very long vectors of social communication skills [our host, here, for instance] who may not be the longest line on the graph for either math or words. Still: genius.) But whether two or three or more “vectors” of intelligence, I think it fair that the education system be tasked to identify those with particular aptitudes early. And then our system should allow and assist thosei identified to refine and develop exceptional gifts, while being relatively tolerant of such persons’ other, more typical, endowments. We needn’t expect Godel to write poetry. We needn’t expect Robert Frost to master Bayesian statistics.

But coming back to my original point, I’m not aware of any comparable “gate” in the primary and secondary “core curriculum” that filters out or marks for advancement students by aptitudes other than math.

Well, I take it back. There is the athletic program.

(A fellow named Kenny Felder tried to leave a comment but it got deleted, so he emailed it to me. Here it is.)

Since I read your blog, I’ve told a lot of people about it very excitedly. Because–and this is one of my favorite things when it happens to me–I had never quite thought about it that way, and yet as soon as I read it, I thought “That’s the right way to frame this question.” It doesn’t start with “We need math, and therefore…” It starts with “We need selection criteria, and therefore…”

I wrote my own little blog–probably about the same length as yours–touching on some of the same issues. If you take the time to read it, I would love to hear your reactions. It’s here:

http://www.felderbooks.com/kennyessays/fantasyschool.html

But hiding behind your discussion is an amazingly common misperception. A lot of my students (and their parents) think “intelligence” is synonymous with “math ability.” That works out OK for me because I’m a math teacher, I guess. But it’s also dead wrong. Some of the smartest, most insightful, most creative, and even most logical people I know are not good at math. It’s not their “thing” for one reason or another. So that causes a huge problem if math is the primary (or sometimes the only) gatekeeper.

Suppose I were starting a university or a think tank or a club, and I wanted to be painfully selective, allowing in only the cream of the intellectual crop. My gatekeeper would be–to use a good math/logic phrase–a giant OR statement. You can join my club if you can solve partial differential equations, OR you can find the logical inconsistencies in a Supreme Court argument, OR you can compose and arrange your own opera, OR you can beautifully translate Shakespeare into French, OR…OR…OR. You see what I mean? We can define intelligence very broadly, and avoid excluding brilliant people who perhaps never got past long division.

In my classroom I emphasize mathematics as a way of thinking logically and coherently, rather than a set of rules. It should always be taught that way, and it often isn’t, and that’s a shame. But even allowing for that, there are still going to be plenty of people whose intelligence and logical abilities come out in other arenas. There is no reason to make those people slog their way through logarithms, except as a gory national hazing ritual: “I survived it, and you will too” often seems to be the only motivation.

-Kenny Felder

(Ben here again: In my post I tried to sidestep the question of whether math attainment is an *accurate* measure of anything, addressing it only with the line “We all know it’s not a perfect indicator. But admissions and HR departments seem to find it useful anyway.” Kenny’s discussion is more thorough and I generally agree with it.)